Integral de $$$\frac{y^{3}}{x^{4}}$$$ em relação a $$$x$$$

Encontre $$$\int \frac{y^{3}}{x^{4}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=y^{3}$$$ e $$$f{\left(x \right)} = \frac{1}{x^{4}}$$$:

$${\color{red}{\int{\frac{y^{3}}{x^{4}} d x}}} = {\color{red}{y^{3} \int{\frac{1}{x^{4}} d x}}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-4$$$:

$$y^{3} {\color{red}{\int{\frac{1}{x^{4}} d x}}}=y^{3} {\color{red}{\int{x^{-4} d x}}}=y^{3} {\color{red}{\frac{x^{-4 + 1}}{-4 + 1}}}=y^{3} {\color{red}{\left(- \frac{x^{-3}}{3}\right)}}=y^{3} {\color{red}{\left(- \frac{1}{3 x^{3}}\right)}}$$

Portanto,

$$\int{\frac{y^{3}}{x^{4}} d x} = - \frac{y^{3}}{3 x^{3}}$$

Adicione a constante de integração:

$$\int{\frac{y^{3}}{x^{4}} d x} = - \frac{y^{3}}{3 x^{3}}+C$$

$$$\int \frac{y^{3}}{x^{4}}\, dx = - \frac{y^{3}}{3 x^{3}} + C$$$A